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A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks Out of Thermal Equilibrium

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Abstract

We continue the investigation, started in Jakšić et al. in (J. Stat. Phys. 166:926–1015, 2017), of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function.

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Notes

  1. see [4], and also, in view of the now well understood connection with the thermodynamic formalism, [49, Proposition 5.3.2, Equ. (3.9)].

  2. The integral there is to be taken in Itô’s sense.

  3. \(X_\xi \) is maximal whenever \(X\le X_\xi \) for all self-adjoint X such that \(\mathscr {R}_\xi (X)=0\).

  4. Here, we used the fact that the steady state covariance M satisfies \(M^{-1}=\theta X_{\vartheta ^{-1}}\theta \).

  5. The lineality space of a convex set \(\mathscr {C}\subset \mathbb {R}^n\) is the set of vectors \(y\in \mathbb {R}^n\) such that \(x+\lambda y\in \mathscr {C}\) for all \(x\in \mathscr {C}\) and all \(\lambda \in \mathbb {R}\), see [47]

  6. Recall that M, given in (2.8), is the covariance of the invariant measure \(\mu \).

  7. In Figures 6 and 7 the set \(\partial \mathscr {S}\) is mapped to the closed unit disk by first mapping \(\partial \mathscr {S}\) to the unit sphere and then mapping the point with spherical coordinates \((\varphi ,\theta )\in [0,2\pi ]\times [0,\pi ]\) on this sphere to the point \(\frac{\theta }{\pi }(\cos \varphi ,\sin \varphi )\) of the plane.

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Acknowledgements

This research was supported by the Agence Nationale de la Recherche (ANR) through the grant NONSTOPS (ANR-17-CE40-0006) and the CNRS collaboration grant Fluctuation theorems in stochastic systems. The work of C.-A.P. has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the ANR. Parts of this work were performed during the visits of M.D. and M.H. at the University of Toulon and of C.-A.P. at the University of Sfax. We thank the CPT, the University of Toulon and the Mathematics Department of Sfax for their hospitality and support. M.H. and C.-A.P. are also grateful to the Centre de Recherches Mathématiques de l’Université de Montréal for its hospitality and the Simons foundation and CNRS for their support during their stay in Montréal in the fall 2018.

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  1. Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia

    Mondher Damak & Mayssa Hammami

  2. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Mayssa Hammami & Claude-Alain Pillet

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  1. Mondher Damak
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Damak, M., Hammami, M. & Pillet, CA. A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks Out of Thermal Equilibrium. J Stat Phys 180, 263–296 (2020). https://doi.org/10.1007/s10955-019-02398-x

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